Conventional CT employs a technique for obtaining cross sectional slices of an object from planar parallel or fan beam irradiation of an object. The technique is primarily utilized in medical and industrial diagnostics. Traditional image reconstruction techniques have been predominantly two dimensional. In three dimensions, an undistorted image of an object can be mathematically reconstructed in an exact manner by back projecting a parallel beam which has been attenuated after passing through an object using an inverse transform based on the Fourier Slice Theorem. The use of a parallel beam source and a flat two dimensional detector geometrically simplifies reconstruction but complicates speed and ease of data collection.
Image reconstruction can be mathematically accomplished for a 3D cone beam source by an inverse Radon transform using suitable planar integrals. These planar integrals are computed from detector integrals which utilize measured cone beam projection data i.e. the detected attenuated intensity representative of the density distributions of the irradiated object. The use of a 3D cone beam source expedites data acquisition, but complicates geometrical considerations when used with a conventional flat array detector.
In two dimensions, the analog of cone beam source geometry is illustrated by tan beam geometry. For the case of two dimensional fan beam geometry, the detector integrals are equivalent to the Radon transform of the object. Unlike the two dimensional case, a direct Radon inversion of three dimensional cone beam data from a cone beam source is not possible. Before the inverse Radon transform can be undertaken in three dimensions, the cone beam detector integrals must be reconfigured into planar integrals suitable for direct inverse Radon transformation. Due to the limitations of direct inversion, three dimensional CT imaging has traditionally involved stacking slices representative of the density distribution through the object obtained from various parallel or fan beam attenuation projections. Each projection is associated with a particular view angle or illumination configuration of source and detector relative to the object. A Radon data set is generally acquired by either rotating a source and detector, fixed relative to one another, around an object taking projections as the object is scanned; or alternatively, rotating the object between the fixed source and detector. Such a Radon data set comprises plurality of discrete data points corresponding to projected attenuated intensity at discrete grid points of an array detector.
Three dimensional Radon inversion is addressed using a two step approach to perform an inverse Radon transform on planar integrals representing cone beam data obtained on a plurality of coaxial planes in Radon space. The first step involves calculating from the planar integrals a two dimensional projection image of the object on each of a plurality of coaxial planes; while the second step involves defining normal slices through these coaxial planes which a two dimensional reconstruction of each slice is obtained. In this slice by slice way, the reconstruction algorithms operate on the plurality of planar integrals to produce a three dimensional image of the object.
The acquired Radon data set is complete only if it provides sufficient Radon data at every necessary point Radon space, i.e. Radon space must be sufficiently filled with data over the region of support in Radon space which corresponds to that region of space occupied by the object in object space. Sufficient filling of Radon space by scanning along a suitable trajectory is necessary for exact image reconstruction. Furthermore, if the detector integral space is filled over the region of support for the object, the Radon data set is said to be complete. Bruce D. Smith in an article entitled "Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods," IEEE Trans. Med. Imag., MI-4 (1985) 14, describes a cone beam data set as `complete` if each plane passing through the object cuts the scanning trajectory in at least one point. This criterion assumes that the detector is fixed relative to the source and that the entire object can be scanned in a continuous manner within the field of view of the source beam. Depending on the scanning trajectory employed to obtain the cone beam projection data, the acquired data set in Radon space may or may not be complete. Data collected using a commonly adopted single circular scan is incomplete and artifacts may accordingly be introduced into the reconstructed image. Dual parallel circular scanning trajectories have been shown to reduce data set incompleteness. A circular square wave scanning trajectory, as well as, dual mutually perpendicular circular scanning trajectories provide a complete Radon data set for exact image reconstruction having been shown to satisfy the completeness criterion as articulated by Smith. More recently, Bruce D. Smith in article entitled "Cone-beam Tomography: Recent Advances and a Tutorial Review", Optical Engineering, Vol. 29, No. 5, pp. 524-534, May 1990, mentions several complete scanning trajectories.
The volume of Radon space must not only be filled in a sufficiently dense manner to accommodate unique reconstruction from a complete data set; but, Radon data must also be acquired in a substantially uniform manner to reflect consistency in the inversion process. This too is accomplished by employing suitable sampling about an appropriate scanning trajectory.
Utilizing an incomplete and/or non-uniform data set for image reconstruction by Radon inversion introduces artifacts which compromise image quality and may render the image inadequate for medical or industrial diagnostic use. The density of acquired Radon data, the distribution of this density, and the volume of data obtained all contribute to the accuracy and efficiency of image reconstruction. Generally, choice of scanning trajectory is the only major consideration regarding data acquisition. Radon data is typically acquired throughout the whole of Radon space, without regard to any apriori selectivity as to what data is necessary based on the shape and character of the object being scanned. Proper dana acquisition typically involves acquiring a complete data set throughout all of Radon space having sufficiently dense information to accurately reconstruct the image in a uniform manner. Sampling conventions are generally only concerned with location, i.e. where to scan for data, and step size, i.e. how far apart data should be sampled along a scan path.
It would be desireable for data acquisition and sampling to proceed in a more computationally expedient manner wherein data is acquired and/or retained nor only a necessary volume of Radon space. According to this invention, the necessary volume of Radon space to be filled is predetermined in an apriori manner accorded by the shape of the object and implemented into a data collect ion and/or retention scheme. Such apriori selectivity can reduce unnecessary, redundant computational requirement by eliminating the need to fill regions of Radon space with unnecessary data. The collect of only necessary and sufficient Radon data for exact reconstruction without unnecessarily over-burdening computational requirements has not been addressed in the prior art. Using apriori information relating to the shape of the object irradiated within the field of view of the irradiating source eliminates unnecessary computational requirements by simply disregarding that data which is not necessary.